History of logarithms

The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer.

Johannes Kepler dedicated his 1620 Ephemerides to Napier, stating that the invention of logarithms was the central idea that enabled him to discover the third law of planetary motion.

Michael J. Bradley, Mathematics Frontiers: 1950 to Present (2006)

Although the slide rule has largely given way to the pocket calculator... the pedagogical value of making one's own [slide or other] rules of various kinds remains. Napier's rods are easily made by schoolchildren and the historical route by which the modern slide rule evolved can be followed through with advantage. If lattice multiplication has been taught at some stage, the principles of Napier's rods will already be understood. Many different kinds of graduated rule can be experimented with, including those of arithmetic scales (to be used for addition and subtraction) and those with geometric scales (for multiplication and division). It is not necessary to mention the word 'logarithm'; it is sufficient to introduce arithmetic and geometric series and to utilize the rules themselves in order to introduce the principles of logarithms.

Graham Flegg, Numbers: Their History and Meaning (1983)

The learned calculators, about the close of the 16th, and beginning of the 17th century, finding the operations of multiplication and division by very long numbers, of 7 or 8 places of figures, which they had frequently occasion to perform, in resolving problems relating to geography and astronomy, to be exceedingly troublesome, set themselves to consider, whether it was not possible to find some method of lessening this labour, by substituting other easier operations in their stead. In pursuit of this object, they reflected, that since, in every multiplication by a whole number, the ratio, or proportion, of the product to the multiplicand, is the same as the ratio of the multiplier to unity, it will follow that the ratio of the product to unity (which, according to Euclid's definition of compound ratios, is compounded of the ratios of the said product to the multiplicand and of the multiplicand to unity) must be equal to the sum of the two ratios of the multiplier to unity and of the multiplicand to unity. ...
And therefore they thought these artificial numbers, which thus represent, or are proportional to, the magnitudes of the ratios of the natural numbers to unity, might not improperly be called the Logarithms of those ratios, since they express the numbers of smaller ratios of which they are composed. And then, for the sake of brevity, they called them the Logarithms of the said natural numbers themselves, which are the antecedents of the said ratios to unity, of which they are in truth the representatives.

Jost Burgi, a Swiss clockmaker and mathematician, invented logarithms independently of Napier and Briggs, although it is not clear when he started work on them. Some historians have suggested that Burgi may have invented logarithms earlier than Napier, but his work was not published until 1620, when the German mathematician and astronomer Johannes Kepler asked him to do so. ...six years after the publication of Napier's work.

In the seventeenth century, perhaps the greatest of all for the development of mathematics, there appeared a work which in the history of British science can be place second only to Sir Isaac Newton's monumental Pincipia. In 1614, John Napier of Merchiston issued his Mirifici Logarithmorum Canonis Descriptio, ("A Description of the Admirable Table of Logarithms"), the first treatise on logarithms. To Napier, who also invented the decimal point, we are indebted for an invention which is as important to mathematics as Arabic numerals, the concept of zero, and the principle of positional notation. Without these, mathematics would probably not have advanced much beyond the stage to which it had been brought two thousand years ago. Without logarithms the computations accomplished daily with ease by every mathematical tyro would tax the energies of the greatest mathematicians.

Johannes Kepler provided more accurate values for the Napier series with the aid of successive proportions between two given terms. In the Tabulae Rudolphinae (1627), he was the first to divide a table of logarithms into numerical and trigonometric parts.

As contrasted with an absolute number, a logistic number represented measurement. ...Kepler welcomed the invention of logarithms as an ingenious device to facilitate laborious computations. Since he was concerned principally with astronomical computations involving sexagesimal fractions of the degree and of the hour, logistic logarithms were of prime importance to him. ...Kepler's logarithms were based on proportion, as he made clear in the following definition of a logarithm in his Thousand Logarithms: "Express the measurement of every proportion between 1000 and a number smaller than 1000... by a number which is placed alongside this smaller number in the Thousand and which is called its logarithm, that is, the number (arithmos) indicating the proportion (logos) which that number, to which the logarithm is attached, bears to 1000."

It [the Rudolphine Tables] was only the third new set of planetary tables in European history. And whereas Copernicus's and Ptolemy's tables were more or less equally accurate, Kepler's were some 50 times more so. Within a few years, it was possible to pinpoint the time of transit of Mercury across the face of the sun so that it was possible to observe it in transit for the first time in human history. Of course, Kepler's theories were more difficult, especially since he had incorporated logarithms, which had only been invented a few years earlier. Much of the book, therefore, was made up of explanatory text that told the reader how to use the tables.

: From Its Institution to the End of the Eighteenth Century by Thomas Thomson, Book II. Of Mathematics

The next improvement in mathematics, which we have to mention, is the introduction of logarithms, those numbers so important by diminishing the labour of tedious calculations, and which play so conspicuous a part in the transcendental analysis. For this admirable discovery we are indebted to John Napier, Baron of Merchiston, near Edinburgh. ...Napier seems to have turned the bent of his genius towards the discovery of methods to facilitate and abridge trigonometrical calculations; and various contrivances were proposed by him in succession, all remarkable for their ingenuity. The last and most memorable of all was his discovery of logarithms.

There is a story told by Mr. Wood, but it does not appear entitled to any attention, that one Dr. Craig, a Scotchman, coming out of Denmark into his own country, called upon John Napier... and told him of a new invention in Denmark, by Longomontanus, to save tedious multiplications and divisions in astronomical calculations. Napier being solicitous to know further of him of this matter, he could give no other account of it than that it was by proportional numbers; which hint Napier taking, desired him, at his return, to call upon him again. Craig, after an interval of some weeks, did so, and Napier then showed him a rude draught of what he called canon mirabilis logarithmorum. Had there been any truth in such a story, we may be sure that Longomontanus and the Danes would not have abstained from laying their claim to so admirable a discovery.

Napier has also been considered as having been anticipated in his invention by Stifels, and by Juste Byrge, two German mathematicians; but these allegations originating from jealousy, or from national partiality, are entitled to no attention whatever, and Napier's claims have for many years been allowed by the universal consent of all mankind.

The logarithms which first presented themselves to Napier were those at present known by the name of hyperbolic logarithms. But it afterwards occurred to him that logarithms, similar to those in our modern tables, in which the logarithm of 1 is 0; that of 10, 1; that of 100, 2; &c., would be more convenient. But he died, in 1618, before he had time to put his new plan in execution; but not till he had explained its nature to Mr. Henry Briggs, Gresham Professor of Mathematics, who had seen at once all the importance of logarithms, and had early devoted himself to bring them to perfection.

Henry Briggs... applied himself chiefly to the study of mathematics. ...As soon as the Napierian discovery of logarithms was announced, he made two successive journeys into Scotland, to confer with the discoverer himself, and settle plans for the calculation and construction of logarithmic tables. An account of the nature and properties of logarithms was published at Edinburgh, in 1618, by Robert Napier, the son of the great discoverer, under the following title: Mirifici Logarithmorum Canonis constructio et eorum ad Naturales ipsorum Numeros Habitudines una cum Appendice de alia caque prestantiori Logarithmorum Specie condenda, &c. &c. This book had been written, and was ready for the press, when John Napier, the inventor of logarithms, was prevented from publishing it by his death. The same year Briggs published a table of the logarithms of the first 1,000 natural numbers, under the title of Logarithmorum Chilias prima. In 1624, he published, under the title of Arithmetica Logarithmica, the logarithms of all numbers from 1 to 20,000 and from 90,000 to 100,000, calculated to 14 decimal places.

Briggs was assisted in his calculations by Gunter... the contriver of the graduated rule which passes under his name. He calculated the logarithms of the sines and tangents, and published a table of them in 1620, entitled, Canon of Triangles. Briggs had made considerable progress in a table of sines and tangents, calculated to 100 parts of a degree, (for he wished to introduce the decimal notation into trigonometry) but died, in 1630, before he had completed it. It was finished by Henry Gellibrand... and he published it in 1633, under the title of Trigonometria Britannica.

One of the first persons on the Continent who properly appreciated the importance of logarithms, was Kepler. He published a work on the subject in 1624, in which he simplified the theory considerably, and developed the views of Napier with great sagacity and simplicity.

The invention of logarithms, without which many of the numerical calculations which have constantly to be made would be practically impossible, was due to Napier of Merchiston. ...he had privately communicated a summary of his results to Tycho Brahe as early as 1594. ...Napier explains the nature of logarithms by comparison between corresponding terms of an arithmetical and geometrical progression. ...it is the first valuable contribution to the progress of mathematics which was made by any British writer. The method by which logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619... Napier had determined to change the base to one which was a power of 10, but died before he could effect it.

The rapid recognition throughout Europe of the advantages of using logarithms in practical calculations was mainly due to Briggs, who was one of the earliest to recognize the value of Napier's invention. Briggs at once realized that the base to which Napier's logarithms were calculated was inconvenient; he accordingly visited Napier in 1616, and urged the change to a decimal base, which was recognized by Napier as an improvement. On his return Briggs immediately set to work to calculate tables to a decimal base, and in 1617 he brought out a table...

J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers... published in 1620.

In [1620] a table of the logarithms... of sines and tangents of angles in the first quadrant was brought out by Edmund Gunter... Four years later [he] introduced a "line of numbers," which provided a mechanical method for finding the product of two numbers: this was the precursor of the slide-rule, first described by Oughtred in 1632.

In 1624, Briggs published tables of the logarithms of some additional numbers and of various trigonometrical functions. ...The calculation of 70,000 numbers which had been omitted by Briggs was performed by Adrian Vlacq and published in 1628: with this addition the table gave logarithms of numbers from 1 to 101,000.

The Arithmetica Logarithmica of Briggs and Vlacq are substantially the same as existing tables: parts have at different times been recalculated but no tables of an equal range and fulness entirely founded on fresh computations have been published since. These tables were supplemented by Brigg's Trigonometrica Britannica, which contains tables not only of the logarithms of the trigonometrical functions, but also of their natural values... published posthumously in 1633.

A table of logarithms to the base e... and of the sines, tangents, and secants of angles in the first quadrant was published by John Speidell... as early as 1619, but... these were not as useful in practical calculations as those to the base 10.

The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the seventeenth century was admirably timed, for Kepler was then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the Renaissance German mathematicians had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the calculator. It is no exaggeration to say that the invention of logarithms "by shortening the labours doubled the life of the astronomer."

Logarithms were invented by John Napier... It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers.

What... was Napier's line of thought? ...Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines...

It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases.

Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by ArchimedesStifel and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables.

Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute.

The theory of natural ("hyperbolic") logarithms apparently first suggested itself to mathematicians engaged in the mensuration of spaces between the hyperbola and its asymptotes. About a quarter of a century later, in 1695, Edmund Halley discarded geometrical figures and published a remarkable article containing a purely arithmetical theory of logarithms. In this original and meritorious investigation he lays great stress upon what we now call the "modulus". By Napier's logarithms Halley understands those which give Briggs's logarithms when divided by 2.302 585 or when multiplied by 0.43429448. From this statement it appears that Halley considered Napier's logarithms to be identical with natural logarithms, and we must look upon him as one of the first (perhaps the first) to commit this error. That the two systems are not identical is shown by the following formula:

The confusion marked in the writings of Halley and Saverien spread among French writers. Montuclu, the great mathematical historian of the eighteenth century, made the same mistake; Bossut helped to perpetuate the error.

In England Charles Hutton, who in 1785, published the first edition of his Mathematical Tables (which includes an elaborate and in many respects excellent history of logarithms) describes Napier's logarithms correctly, but subsequently he speaks of "the right-angled hyperbola, the side of whose square inscribed at the vertex is 1, gives "Napier's logarithms".

De Morgan carefully explains the difference between Napier's and natural logarithms in the article "Tables" in the English Cyclopaedia but in De Morgan's Budget of Paradoxes (р. 70) Günther has found a passage which is inaccurate.

It is a pleasure to find that Kästner presents the subject in a way free of error. In his Geschichte he refers to an article, which he had written, setting forth the exact relation between the two systems. Nevertheless the misconception became prevalent in Germany also.

Proceeding to... the earliest publication of tables of natural logarithms... John Speidell... in 1619 brought out his New Logarithmes, only five years after Napier's publication of the Descriptio. Speidell's book received little attention, either during his life-time or since. It would seem as if the earliest publication of a table of natural logarithms should be mentioned in histories of mathematics, but so far as I know, no general history by a German, French, or British author, takes notice of Speidell. ..However, Speidell's New Logarithmes has been described in at least three special historical articles. Hutton speaks of it in the "Introduction" to his Tables; Augustus De Morgan makes a careful study of his book in the article "Tables" in the English Cyclopaedia; a J. W. L. Glaisher gives a brief account of Speidell's work in the report on "Tables" in the British Association Report, 1873...

Speidell's... sole object was to simplify matters for persons unacquainted with the use of negative quantities. ...Speidell did not advance a new theory. He simply aimed to make all the logarithms in his table positive.

The word "logarithm" means "ratio number" and was an afterthought with Napier. He first used the expression "artificial number," but before he announced his discovery he adopted the name by which it is now known.

Briggs introduced (1624) the word "mantissa." ...originally meaning an addition, a makeweight, or something of minor value, and was written mantisa. In the 16th century it came to be written mantissa and to mean "appendix"... The term "characteristic" was suggested by Briggs (1624) and is used in the 1628 edition of Vlacq.

Napier worked at least twenty years upon the theory. His idea was to simplify multiplications involving sines, and it was a later thought that included other operations, applying logarithms to numbers in general. He may have been led to his discovery by the relation

for, as Lord Moulton says, in no other way can we "conceive that the man to whom so bold an idea occurred should we have so needlessly and so aimlessly restricted himself to sines in his work, instead of regarding it as applicable to numbers in general."

Napier published his Descriptio of the table of logarithms in 1614. This was at once translated into English by Edward Wright, but with the logarithms contracted by one figure.

In Napier's time sin φ was a line, not a ratio. The radius was called the sinus totus, and when this was equal to unity the length of the sine was simply stated as sinφ. If r was not unity, the length was r sinφ. With this statement we may consider Napier's definition of a logarithm:

The logarithme therefore of any sine is a number very neerely expressing the line, which increased equally in the meane time, whiles the line of the whole sine decreased proportionally into that sine, both motions being equal-timed, and the beginning equally swift.

From this it follows that the logarithm of the sinus totus is zero. Napier saw later that it was better to take log 1 = 0.

Napier's logarithms are not those of the so-called Naperian, or hyperbolic, system, but are connected with this system by the relation

Henry Briggs... was one of the first to appreciate the work of Napier. Upon reading the Descriptio he wrote

Naper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.

Briggs's Aritmetica Logarithmica the preface... contains the following statement by the author...

That these logarithms differ from those which that illustrious man, the Baron of Merchiston published in his Canon Mirificus must not surprise you. For I myself, when expounding their doctrine publicly in London to my auditors in Gresham College, remarked that it would be much more convenient that 0 should be kept for the logarithm of the whole sine (as in the Canon Mirificus)... And concerning that matter I wrote immediately to the author himself; and as soon as... permitted I journeyed to Edinburgh, where, being most hospitably received by him, I lingered for a whole month. But as we talked over the change in logarithms he said that he had for some time been of the same opinion and had wished to accomplish it. ...He was of the opinion that... 0 should be the logarithm of unity.

The real value of the proposition made by Briggs at this time was that he considered the values of log 10na, for all values of n. The relation between the two systems as they first stood were as follows:

The first table of logarithms of trigonometric functions to the base 10 was made by Gunter.

In the 1618 edition of Edward Wright's translation of the Descriptio there is printed an appendix, probably written by Oughtred, in which there is an equivalent of the statement that loge10 = 2.302584, thus recognizing the base e. Two years later John Speidell published his New Logarithmes, also using this base. He stated substantially that

By the middle of the seventeenth century, logarithms had found their way into elementary arithmetics, as seen in Hartwell's (1646) edition of Recorde's Ground of Artes, where it is said that "for the extraction of all roots, the table of Logarithms set forth by M. Briggs are most excellent, and ready."

It is evident that

23⋅24=27,{\displaystyle 2^{3}\cdot 2^{4}=2^{7},}

(22)3=26,{\displaystyle (2^{2})^{3}=2^{6},}

27:23=24,{\displaystyle 2^{7}:2^{3}=2^{4},}

(24)12=22,{\displaystyle (2^{4})^{\frac {1}{2}}=2^{2},}

which are the fundamental laws of logarithms. Most writers refer to Stifel as the first to set forth these basal laws... but he was by no means the first... Probably the best statements concerning them which appeared in the 15th century were those of Chuquet in Le Triparty en la Science des Nombres... Chuquet expressed very clearly the relations

aman=am+n{\displaystyle a^{m}a^{n}=a^{m+n}}

and

(am)n=amn{\displaystyle (a^{m})^{n}=a^{mn}}

Stifel... in the Arithmetica Integra of 1544 ...refers several times to the laws of exponents. At first he uses the series

0

1

2

3

4

5

6

7

8

1

2

4

8

16

32

64

128

256

distinctly calling the upper numbers exponents, and saying that the exponents of the factors are added to produce the exponent of the product and subtracted to produce the exponent of the quotient. Moreover, he expressly lays down four laws, namely, that addition in arithmetic progression corresponds to multiplication in geometric progression, that subtraction corresponds to division, multiplication to the finding of powers, and division to the extracting of roots. Furthermore, Stifel not only set forth the laws for positive exponents but also saw the great importance of considering the negative exponents of the base which he selected, using the series

-3

-2

-1

0

1

2

3

4

5

6

1/8

1/4

1/2

1

2

4

8

16

32

64

and making the significant remark: I might write a whole book concerning the marvellous things relating to numbers but I must refrain and leave these things with eyes closed." What these mysteries were we can only conjecture.

The theory was again given by [Pierre] Forcadel (1565), with a statement that the idea was due to Archimedes, that it was to be found in Euclid, and that Gemma Frisius had written upon it.

When... Schoner came to write his commentary on the work of Ramus, in 1586, a decided advance was made, for not only did he give the usual series for positive exponents, but, like Stifel, he used the geometric progressions with fractions as well, although... not with negative exponents.

In general the German writers were in the lead. ...particularly ...Simon Jacob (1565), who followed Stifel closely, recognizing all four laws, and... influencing Jobst Bürgi. These writers did not use the general exponents essential to logarithms, but the recognition of the four laws is significant.

In 1620 Jobst Bürgi published his Progress Tabulen... he was influenced by Simon Jacob's work. The tables... are simply lists of antilogarithms with base 1.0001. ...none seem to be later than 1610, so that he probably developed his theory independently of Napier. ...he approached the subject algebraically, as Napier approached it geometrically.